The extreme vertices of the power graph of a group

For a fixed finite group G, the power graph of G was defined to be the simple graph Γ(G) whose vertex set V(Γ(G))=G, and edge set E(Γ(G))={xy: either x=yn or y=xn for some integer n}. In this paper the extreme vertices of the power graph of abelian groups, dihedral groups and dicyclic groups have been characterized.

For a fixed finite group , the power graph of was defined to be the simple graph Γ( ) whose vertex set (Γ( )) = , and edge set (Γ( )) = { ∶ either = or = for some integer }. In this paper the extreme vertices of the power graph of abelian groups, dihedral groups and dicyclic groups have been characterized.

Introduction
For a fixed finite group , the directed power graph of , ⃖ ⃗ Γ( ), was defined by Kelarev et al. (2001), to be the digraph whose vertex set is the elements of the finite group and there is an arc from to if and only if ≠ and ⟨ ⟩ ⊆ ⟨ ⟩, that is = for some integer . Note here that digons (or bidirectional arcs) will appear if and only if ⟨ ⟩ = ⟨ ⟩. The underlying graph of power graph of was first studied by Chakrabarty et al. (2009), it was denoted by Γ( ). To be more clear the underlying graph of the power graph of is the graph with vertex set and two different vertices ≠ are adjacent if and only if one of ⟨ ⟩ and ⟨ ⟩ is subset of the other.
Many researchers were attracted to work on both directed and undirected power graphs of a finite group . For example Cameron and Ghosh (2011) proved that if power graphs of two finite groups and are isomorphic then and have the same number of elements of each order. Cameron (2010) answered the classical isomorphism question: For two abelian groups 1 and 2 , if Γ( 1 ) and Γ( 1 ) are isomorphic, then 1 and 2 are isomorphic. Furthermore, they proved that for a power graph Γ( ) of a finite group , the automorphism group is the same as that of its power graph if and only if is the Klein 4-group. Also, Curtin and Pourgholi (2013) proved that maximum size of power graphs of finite groups, of same order, can be obtained in the set of the cyclic groups. In fact many graph invariants and properties of power graph and power digraph were also investigated, see Aalipour et al. (2017); Tamizh Chelvam and Sattanathan (2013); Mirzargar et al. (2012); Moghaddamfar et al. (2014); Pourgholi et al. (2015). In re-2 … is the prime decomposition of , then ≅ 1 × 2 ×… where = { ∈ ∶ | | is a power of }. Finally, for two positive integers and , we denote greatest common divisor of and by ( , ). Moreover the Euler's totient function of an integer , ( ), (some times called Euler's phi function) is defined to be the number of positive integers less than and co-prime with . For a graph , the neighborhood of a vertex is defined by ( ) = { ∈ ( ) ∶ ∈ ( )}, the degree of the vertex is defined to be ( ) = | ( )|. For any two vertices and of a connected graph , ( , ) denotes the length of a shortest path between and . Finally, a vertex in a graph G is called an extreme vertex if the subgraph induced by its neighborhood is complete.
In this paper we investigate the extreme vertices of the power graph of a finite group. We find the extreme vertices of the power graph of finite abelian groups, dihedral groups and dicyclic groups. The abstract concept of convexity and extreme points concept were introduced and investigated in the fifties of last century. These concepts have been extended to graph theory. In fact extreme points play an essential role in studying abstract convexity especially in graph theory. For example every geodetic set of a graph must contain its extreme vertices.

Extreme vertices of power graph of groups
We examine the conditions on the elements of a group to be extreme vertices of Γ( ).

Theorem 1. Suppose that is an element of a group . If
divides | |, where and are two distinct primes, then is not an extreme vertex of Γ( ).
Proof. Observe that ⟨ ⟩ contains an element of order and an element of order , say these elements are and . The two vertices and are adjacent to in Γ( ) but they are not adjacent in Γ( ). Thus is not an extreme vertex of Γ( ). □ According to the previous theorem, the candidates for extreme vertices in Γ( ) are elements in of prime power order. We state this result in the following corollary.

Corollary 1. Let be an element of a group . If is an extreme vertex
where is a prime number and is a non-negative integer.
In the following results, we examine some cases where elements of prime power order are not extreme vertices.
Theorem 2. Let , and be distinct prime numbers and be an element of a group with | | = , > 0. Suppose that contains an element of order and an element of order with ∈ ⟨ ⟩ and ∈ ⟨ ⟩. Then is not an extreme vertex of Γ( ).
Proof. Since ∈ ⟨ ⟩ and ∈ ⟨ ⟩, then the vertex is adjacent to both of the vertices and in Γ( ). Since | | = , | | = and and are distinct primes, then the vertices and are not adjacent in Γ( ). Therefore is not an extreme vertex of Γ( ). □ Theorem 3. Let and be two distinct prime numbers and be an element of a group with | | = , > 0. Suppose that contains an element of order , > , and an element of order , ≤ < , with ∈ ⟨ ⟩ and ∈ ⟨ ⟩. Then is not an extreme vertex of Γ( ).
Proof. It is similar to the proof of Theorem 2. □ Theorem 4. Let , and be distinct prime numbers and be an element of a group with | | = , > 0. Suppose that contains an element of order such that ∈ ⟨ ⟩. Then is not an extreme vertex of Γ( ).
Proof. Since ∈ ⟨ ⟩, then the vertex is adjacent to in Γ( ). We have | | = and | | = . Note that the two vertices and are adjacent to the vertex but they are not adjacent to each other in Γ( ). Therefore is not an extreme vertex of Γ( ). □

Extreme vertices of power graphs of Abelian groups
Now, let us look at the extreme vertices of Γ( ) where is an abelian group. The following theorem shows that Γ( ) has no extreme vertices for many abelian groups.
Theorem 5. Let , and be distinct prime numbers and be an abelian group. If divides the order of then Γ( ) has no extreme vertices.
Proof. According to Corollary 1, an element ∈ is a possible extreme vertex if its order is of prime power. Suppose that ∈ and | | = 1 where 1 is a prime number. The prime number 1 is not equal to at least two of the primes , and . Without loss of generality we can assume that 1 ≠ and 1 ≠ . Since and are primes, then has an element of order and an element of order , say these elements are and , respectively. Since is abelian, then and are elements of order 1 and 1 , respectively. Using Theorem 2, is not an extreme vertex of Γ( ) and thus Γ( ) has no extreme vertices. □ Let be an abelian group. According to previous theorem Γ( ) can have extreme vertices only if | | = where and are prime numbers. We have the following results for cyclic groups of order .
Theorem 6. Let and be two distinct prime numbers and be a cyclic group of order where and are positive integers. If ∈ and | | = where < , then is not an extreme vertex of Γ( ).

Proof. Suppose that
where is an element of order . Consider the two elements = and = − where is an element of order in . Since gcd( , ) = 1, then there exists a positive integer such ≡ 1 modulo . Therefore, there exists an integer such that The last equality follows from the fact orem 3 to get is not an extreme vertex of Γ( ). □ Theorem 7. Let and be two distinct prime numbers and be a cyclic group of order where and are positive integers. If ∈ and | | = , then is an extreme vertex of Γ( ).
Proof. Suppose that ∈ and | | = . Since is a cyclic group of order , then = where is a cyclic group of order and is a cyclic group of order with ∩ = { } ( is the internal direct product of and ). Since | | = = | |, then = ⟨ ⟩ and the vertex is adjacent to all the elements of = ⟨ ⟩, i.e. is adjacent to all elements of order , ≤ . These are all the elements such that | | divides | | and it is clear that all of these elements are mutually adjacent. If is another element that is adjacent to , then | | divides | | and thus | | = for some > 0. Since is cyclic and | | = , then ⟨ ⟩ ⊆ ⟨ ⟩ and thus is adjacent to . Therefore the vertices that are adjacent to are all the elements of order where 0 ≤ ≤ and elements of order where 1 ≤ ≤ . It is clear that elements of order where 0 ≤ ≤ are adjacent to each other. It is easy to check that elements of order where 0 ≤ ≤ are adjacent to elements of order where 1 ≤ ≤ . Suppose that 1 and 2 are elements of order 1 and 2 , respectively with 1 ≤ 2 . Since is cyclic, then ⟨ 1 ⟩ ⊆ ⟨ 2 ⟩ and so 1 and 2 are adjacent in Γ( ). Hence is an extreme vertex of Γ( ). □ We get the same results for elements of order where ≤ . Combining these results with Theorem 1 we get the following corollary.

Corollary 2. Let and be two distinct prime numbers and be a cyclic group of order
where and are positive integers. An element ∈ is an extreme vertex of Γ( ) if and only if | | = or .
According to Theorem 5, Theorem 6 and Theorem 7 we get the following classification for the extreme vertices of the power graph of a cyclic group.
Corollary 3. Let be a cyclic group and , are prime numbers.

Now, we want to look at abelian groups of the form
≥ 1 and ≥ 2 (case = 1 gives cyclic groups and we have discussed cyclic groups before). Suppose that ∈ and is of the form = 1 1 2 2 … 1 1 , where the 's ≥ 0 and 0 < 1 < | 1 |. If one of the 's > 0, then using Theorem 1, is not an extreme vertex of Γ( ). If all 's = 0, then = 1 and using a similar argument to the one before Theorem 8, we get is not an extreme vertex of Γ( ). Thus candidates for extreme vertices are elements of the form = 1 1 2 2 … , where the 's ≥ 0. In the following theorems, we characterize which ones of these elements are extreme vertices.
Suppose that is an abelian group and ≅ ⟨ 1 ⟩ ×⟨ 2 ⟩ ×… ⟨ ⟩ ×⟨ 1 ⟩, where | | = , 1 ≤ ≤ , | 1 | = , 1 ≤ 1 ≤ 2 ≤ ⋯ ≤ , ≥ 1 and ≥ 2. According to previous theorem, the only candidates for extreme vertices in Γ( ) are elements of the form = 1 1 2 2 … such that | | = and = for at least one . Ultimately, we will show that these elements are the extreme vertices of Γ( ). First, we want to find all the elements that are adjacent to where = 1 1 2 2 … such that | | = and = for at least one . To do that we need the following notations and results from Sehgal and Singh (2019). We write + ( ), − ( ) and ± ( ) to denote respectively the out-degree of , the in-degree of and the number of bidirectional edges incident to in the diagraph ⃖ ⃗ Γ( ). Note that the degree of a vertex in Γ( ) equals the sum of the in-degree and out-degree of minus the number of bidirectional edges incident to . It is easy to check that + ( ) = |⟨ ⟩| − 1 = | | − 1 and ± ( ) = (| |) − 1. Thus ( ) = | | − (| |) + − ( ). To determine ( ), we need to count − ( ). In Sehgal and Singh (2019), the authors investigated this problem for abelian groups and gave the following results.
Theorem 11. (Sehgal and Singh (2019)) Let be a group and let and be two normal subgroups of such that | | and | | are relatively prime. If is the internal direct product of the subgroups and , then for an element = of the group where ∈ and ∈ , As mentioned earlier, our objective is to find all the elements that are adjacent to where = 1 1 2 2 … such that | | = ( ≤ ) and = for at least one . The element is in the abelian group = ⟨ 1 ⟩. Observe that = for at least one , and thus we get Therefore, according to Theorem 10 we get − ( ) = (| |) − 1 and thus ( ) = | | − (| |) + − ( ) = | | − 1. Write = ⋅ 1 where ∈ and 1 ∈ . Use Theorem 11 to get + ( ) = ( + ( ) + 1 ) ( Consider the sets = ⟨ ⟩ − { } and = { 1 ∶ gcd( , ) = 1 and 1 ≤ ≤ − 1}. It is easy to check that | ∪ | = (| | − 1) + (| |)(| | − 1) and each element of ∪ is adjacent to . Since ( ) = | ∪ | and each element of ∪ is adjacent to , then the elements of ∪ are precisely the vertices that are adjacent to in Γ( ). We are now in a position to determine the extreme vertices of Γ( ).

Theorem 12. Suppose that is an abelian group and
≥ 1 and ≥ 2. The extreme vertices of Γ( ) are precisely the elements of the form = 1 1 2 2 … such that | | = and = for at least one .
Using a similar argument, we get the following result.

Extreme vertices of power graphs of dihedral and dicyclic groups
In this section, we examine the extreme vertices for the dihedral and dicyclic groups. The dihedral group of order 2 is 2 = ⟨ , ∶ = 2 = 1 and = −1 ⟩.
The characterization of the power graphs of dicyclic groups was given in Chattopadhyay and Panigrahi (2014).